The proof of the vis-viva equation of orbital mechanics found on wikipedia looks, in my opinion, somewhat convoluted and unenlightening. Considering the simplicity and importance of the vis-viva equation, is there a shorter or more insightful derivation?

The vis-viva equation states that for a Kepler orbit of a mass around a central mass $M$, the magnitude of the velocity at any distance $r$ from the center obeys $v^2 = GM(\frac{2}{r}-\frac{1}{a})$, where $a$ is the semi-major axis of the orbit. It is equivalent to saying that the energy of a Kepler orbit with semi-major axis $a$ is $-\frac{GM}{2a}$. This is a very natural generalization of the 'circular' case where the total energy is $-\frac{GM}{2r}$.

Because of the simplicity of the result, I guessed that there could be a relatively straightforward derivation. The derivation on wikipedia involves quite some algebra and, importantly, only holds for elliptical orbits.

  • $\begingroup$ I suggest you should add some more detail, namely the equation itself, and which bit of the derivation you find "unenlightening". As it stands, this question is not posed very well. $\endgroup$ – Flint72 May 12 '14 at 12:27
  • $\begingroup$ The derivation given at Wikipedia only applies when one of the masses is negligibly small. For the general 2-body problem the energy is $ -\frac{GMm}{2a} $, and the specific energy is $ -\frac{G(M+m)}{2a} $. $\endgroup$ – Nick Jul 13 '14 at 5:57

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